3.616 \(\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=172 \[ \frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac{11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^3 x \]

[Out]

-3*a^3*x - (15*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d + (a^3*Cot[c
+ d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x])/(
16*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(2*d)

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Rubi [A]  time = 0.28983, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2872, 3767, 8, 3768, 3770, 2638} \[ \frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac{11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

-3*a^3*x - (15*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d + (a^3*Cot[c
+ d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x])/(
16*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(2*d)

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (-3 a^9+8 a^9 \csc ^2(c+d x)+6 a^9 \csc ^3(c+d x)-6 a^9 \csc ^4(c+d x)-8 a^9 \csc ^5(c+d x)+3 a^9 \csc ^7(c+d x)+a^9 \csc ^8(c+d x)-a^9 \sin (c+d x)\right ) \, dx}{a^6}\\ &=-3 a^3 x+a^3 \int \csc ^8(c+d x) \, dx-a^3 \int \sin (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^7(c+d x) \, dx+\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (6 a^3\right ) \int \csc ^4(c+d x) \, dx+\left (8 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (8 a^3\right ) \int \csc ^5(c+d x) \, dx\\ &=-3 a^3 x+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{d}+\frac{2 a^3 \cot (c+d x) \csc ^3(c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac{1}{2} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx-\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac{\left (6 a^3\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (8 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-3 a^3 x-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac{1}{8} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-3 a^3 x+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac{1}{16} \left (15 a^3\right ) \int \csc (c+d x) \, dx\\ &=-3 a^3 x-\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 1.35453, size = 292, normalized size = 1.7 \[ \frac{a^3 \left (4480 \cos (c+d x)+9984 \tan \left (\frac{1}{2} (c+d x)\right )-9984 \cot \left (\frac{1}{2} (c+d x)\right )-35 \csc ^6\left (\frac{1}{2} (c+d x)\right )+350 \csc ^4\left (\frac{1}{2} (c+d x)\right )-1050 \csc ^2\left (\frac{1}{2} (c+d x)\right )+35 \sec ^6\left (\frac{1}{2} (c+d x)\right )-350 \sec ^4\left (\frac{1}{2} (c+d x)\right )+1050 \sec ^2\left (\frac{1}{2} (c+d x)\right )+4200 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-4200 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-7664 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-\frac{5}{2} \sin (c+d x) \csc ^8\left (\frac{1}{2} (c+d x)\right )-17 \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )+479 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+5 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )+34 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )-13440 c-13440 d x\right )}{4480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(-13440*c - 13440*d*x + 4480*Cos[c + d*x] - 9984*Cot[(c + d*x)/2] - 1050*Csc[(c + d*x)/2]^2 + 350*Csc[(c
+ d*x)/2]^4 - 35*Csc[(c + d*x)/2]^6 - 4200*Log[Cos[(c + d*x)/2]] + 4200*Log[Sin[(c + d*x)/2]] + 1050*Sec[(c +
d*x)/2]^2 - 350*Sec[(c + d*x)/2]^4 + 35*Sec[(c + d*x)/2]^6 - 7664*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 479*Csc[
(c + d*x)/2]^4*Sin[c + d*x] - 17*Csc[(c + d*x)/2]^6*Sin[c + d*x] - (5*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + 998
4*Tan[(c + d*x)/2] + 34*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 5*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(4480*d)

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Maple [A]  time = 0.096, size = 228, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}+{\frac{5\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{16\,d}}+{\frac{15\,{a}^{3}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{15\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{3\,{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}-3\,{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-3\,{a}^{3}x-3\,{\frac{{a}^{3}c}{d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x)

[Out]

-1/8/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7+3/16/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7+3/16*a^3*cos(d*x+c)^5/d+5/16*a^3*cos
(d*x+c)^3/d+15/16*a^3*cos(d*x+c)/d+15/16/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-3/5*a^3*cot(d*x+c)^5/d+a^3*cot(d*x+c)
^3/d-3*a^3*cot(d*x+c)/d-3*a^3*x-3/d*a^3*c-1/2/d*a^3/sin(d*x+c)^6*cos(d*x+c)^7-1/7/d*a^3/sin(d*x+c)^7*cos(d*x+c
)^7

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Maxima [A]  time = 1.60888, size = 315, normalized size = 1.83 \begin{align*} -\frac{224 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} - 35 \, a^{3}{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 70 \, a^{3}{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{160 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/1120*(224*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^3 - 35*a^3*(2*(33*c
os(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1
) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 70*a^3*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(
d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) +
160*a^3/tan(d*x + c)^7)/d

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Fricas [B]  time = 1.22366, size = 871, normalized size = 5.06 \begin{align*} -\frac{4992 \, a^{3} \cos \left (d x + c\right )^{7} - 12992 \, a^{3} \cos \left (d x + c\right )^{5} + 11200 \, a^{3} \cos \left (d x + c\right )^{3} - 3360 \, a^{3} \cos \left (d x + c\right ) + 525 \,{\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 525 \,{\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 70 \,{\left (48 \, a^{3} d x \cos \left (d x + c\right )^{6} - 16 \, a^{3} \cos \left (d x + c\right )^{7} - 144 \, a^{3} d x \cos \left (d x + c\right )^{4} + 33 \, a^{3} \cos \left (d x + c\right )^{5} + 144 \, a^{3} d x \cos \left (d x + c\right )^{2} - 40 \, a^{3} \cos \left (d x + c\right )^{3} - 48 \, a^{3} d x + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/1120*(4992*a^3*cos(d*x + c)^7 - 12992*a^3*cos(d*x + c)^5 + 11200*a^3*cos(d*x + c)^3 - 3360*a^3*cos(d*x + c)
+ 525*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2)*si
n(d*x + c) - 525*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c
) + 1/2)*sin(d*x + c) + 70*(48*a^3*d*x*cos(d*x + c)^6 - 16*a^3*cos(d*x + c)^7 - 144*a^3*d*x*cos(d*x + c)^4 + 3
3*a^3*cos(d*x + c)^5 + 144*a^3*d*x*cos(d*x + c)^2 - 40*a^3*cos(d*x + c)^3 - 48*a^3*d*x + 15*a^3*cos(d*x + c))*
sin(d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**8*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.32594, size = 393, normalized size = 2.28 \begin{align*} \frac{5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 35 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 49 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 245 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 875 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 455 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 13440 \,{\left (d x + c\right )} a^{3} + 4200 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 9065 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{8960 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - \frac{10890 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 9065 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 455 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 875 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 245 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 49 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{4480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/4480*(5*a^3*tan(1/2*d*x + 1/2*c)^7 + 35*a^3*tan(1/2*d*x + 1/2*c)^6 + 49*a^3*tan(1/2*d*x + 1/2*c)^5 - 245*a^3
*tan(1/2*d*x + 1/2*c)^4 - 875*a^3*tan(1/2*d*x + 1/2*c)^3 + 455*a^3*tan(1/2*d*x + 1/2*c)^2 - 13440*(d*x + c)*a^
3 + 4200*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 9065*a^3*tan(1/2*d*x + 1/2*c) + 8960*a^3/(tan(1/2*d*x + 1/2*c)^2
+ 1) - (10890*a^3*tan(1/2*d*x + 1/2*c)^7 + 9065*a^3*tan(1/2*d*x + 1/2*c)^6 + 455*a^3*tan(1/2*d*x + 1/2*c)^5 -
875*a^3*tan(1/2*d*x + 1/2*c)^4 - 245*a^3*tan(1/2*d*x + 1/2*c)^3 + 49*a^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3*tan(
1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^7)/d